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Noneuclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities

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  • Terras, Audrey

Abstract

This is an expository paper. The derivation of the ordinary central limit theorem using the Fourier transform on the real line is reviewed. Harmonic analysis on the Poincaré-Lobatchevsky upper half plane H is sketched. The Fourier inversion formula on H reduces to that for the classical integral transform of F. G. Mehler (1881, Math. Ann. 18, 161-194) and V. A. Fock (1943, Compt. Rend. Acad. Sci. URSS Dokl N. S. 39, 253-256), for example. This result is then used to solve the heat equation on H, producing a non-Euclidean analogue of the density function for the Gaussian or normal distribution on H. The non-Euclidean central limit theorem for rotation invariant distributions on H with an application to the statistics of long transmission lines is also discussed.

Suggested Citation

  • Terras, Audrey, 1984. "Noneuclidean harmonic analysis, the central limit theorem, and long transmission lines with random inhomogeneities," Journal of Multivariate Analysis, Elsevier, vol. 15(2), pages 261-276, October.
    • Handle: RePEc:eee:jmvana:v:15:y:1984:i:2:p:261-276
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